Data-driven trajectory smoothing
Proceedings of the 19th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
Local homology transfer and stratification learning
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Linear-size approximations to the vietoris-rips filtration
Proceedings of the twenty-eighth annual symposium on Computational geometry
New Bounds on the Size of Optimal Meshes
Computer Graphics Forum
Lower bounds for k-distance approximation
Proceedings of the twenty-ninth annual symposium on Computational geometry
Minimax rates of convergence for Wasserstein deconvolution with supersmooth errors in any dimension
Journal of Multivariate Analysis
Information Processing Letters
Noise-adaptive shape reconstruction from raw point sets
SGP '13 Proceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing
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Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (e.g., Betti numbers, normals) of this subset from the approximating point cloud data. It appears that the study of distance functions allows one to address many of these questions successfully. However, one of the main limitations of this framework is that it does not cope well with outliers or with background noise. In this paper, we show how to extend the framework of distance functions to overcome this problem. Replacing compact subsets by measures, we introduce a notion of distance function to a probability distribution in ℝ d . These functions share many properties with classical distance functions, which make them suitable for inference purposes. In particular, by considering appropriate level sets of these distance functions, we show that it is possible to reconstruct offsets of sampled shapes with topological guarantees even in the presence of outliers. Moreover, in settings where empirical measures are considered, these functions can be easily evaluated, making them of particular practical interest.